Abstract
The H ∞ filtering problem for two-dimensional Takagi–Sugeno fuzzy systems described by the Fornasini–Marchesini (FM) model is studied. Attention is focused on the design of an H ∞ fuzzy filter such that the filter error system is asymptotically stable and preserves a guaranteed H ∞ performance. By using basis-dependent Lyapunov functions and adding slack matrix variables, the coupling between the Lyapunov matrix and the system matrices is eliminated. Then, a linear matrix inequality (LMI)-based approach is developed for designing the H ∞ fuzzy filter. Finally, an illustrative example is provided to show the effectiveness of the proposed approach and less conservatism.
Similar content being viewed by others
References
J. An, G. Wen, C. Lin, R. Li, New results on a delay-derivative-dependent fuzzy H ∞ filter design for T-S fuzzy systems. IEEE Trans. Fuzzy Syst. 19(4), 770–779 (2011)
M. Benhayoun, F. Mesquine, A. Benzaouia, Delay-dependent stabilizability of 2D delayed continuous systems with saturating control. Circuits Syst. Signal Process. (2013). doi:10.1007/s00034-013-9585-4
C.W. Chen, J.S.H. Tsai, L.S. Shieh, Two-dimensional discrete-continuous model conversion. Circuits Syst. Signal Process. 18, 565–585 (1999)
J.R. Cui, G.D. Hu, Q. Zhu, Stability and robust stability of 2D discrete stochastic systems. Discrete Dyn. Nat. Soc. 545361, 11 pp. (2011). doi:10.1155/2011/545361
C. De Souza, L. Xie, D. Coutinho, Robust filtering for 2D discrete-time linear systems with convex-bounded parameter uncertainty. Automatica 46(4), 673–681 (2010)
C. Du, L. Xie, H ∞ Control and Filtering of Two-dimensional Systems (Springer, Berlin, 2002)
C. Du, L. Xie, Y. Soh, H ∞ filtering of 2D discrete systems. IEEE Trans. Signal Process. 48(6), 1760–1768 (2000)
C. El-Kasri, A. Hmamed, T. Alvarez, F. Tadeo, Robust H ∞ filtering of 2D Roesser discrete systems: a polynomial approach. Math. Problems Eng. 521675, 15 pp. (2012)
A. Hmamed, A. El Hajjaji, A. Benzaouia, Stabilization of discrete-time 2D T-S fuzzy systems by state feedback control, in 17th Mediterranean Conference on Control and Automation, MED’09, Thessaloniki (2009), pp. 1–6
A. Hmamed, F. Mesquine, F. Tadeo, M. Benhayoun, A. Benzaouia, Stabilization of 2D saturated systems by state feedback control. Multidimens. Syst. Signal Process. 21, 277–292 (2010). doi:10.1007/s11045-010-0107-2
A. Hmamed, M. Alfidi, A. Benzaouia, F. Tadeo, LMI conditions for robust stability of 2D linear discrete-time systems. Math. Probl. Eng. 2008, 356124 (2008)
A. Hmamed, M. Alfidi, A. Benzaouia, F. Tadeo, Robust stabilisation under linear fractional parametric uncertaines of two-dimensional system with Rosser models. Int. J. Sci. Tech. Autom. Control Comput. Eng. (2007). Special Issue. CSC
A. Hmamed, C. El-Kasri, E.H. Tissir, T. Alvarez, F. Tadeo, Robust H ∞ filtering for uncertain 2-D continuous systems with delays. Int. J. Innov. Comput. Inf. Control 9(5), 2167–2183 (2013)
S. Huang, Z. Xiang, Delay-dependent robust H ∞ control for 2-D discrete nonlinear systems with state delays. Multidimens. Syst. Signal Process. (2013). doi:10.1007/s11045-013-0230-y
S. Huang, Z. Xiang, Delay-dependent stability for discrete 2D switched systems with state delays in the Roesser model. Circuits Syst. Signal Process. (2013). doi:10.1007/s00034-013-9600-9
A. Jadbabaie, A reduction in conservatism in stability and L 2 gain analysis of Takagi–Sugeno fuzzy systems via linear matrix inequalities, in Proceedings of the 14th IFAC Triennial World Congress, (1999), pp. 285–289
T. Kaczorek, Two-Dimensional Linear Systems (Springer, Berlin, 1985)
L. Li, W. Wang, X. Li, New approach to H ∞ filtering of two-dimensional T-S fuzzy systems. Int. J. Robust Nonlinear Control (2012). doi:10.1002/rnc.2866
S.K. Nguang, W. Assawinchaichote, H ∞ filtering for fuzzy dynamical systems with D stability constraints. IEEE Trans. Circuits Syst. 50(11), 1503–1508 (2003)
T. Ooba, On stability analysis of 2D systems based on 2D Lyapunov matrix inequalities. IEEE Trans. Circuits Syst. I 47(8), 1263–1265 (2000)
D. Peng, X. Guan, H ∞ filtering of 2D discrete state-delayed systems. Multidimens. Syst. Signal Process. 20(3), 265–284 (2009)
J. Qiu, G. Feng, J. Yang, A new design of delay-dependent robust H ∞ filtering for continuous-time polytopic systems with time-varying delay. Int. J. Robust Nonlinear Control 20, 346–365 (2010). doi:10.1002/rnc.1439
T. Tanaka, T. Hori, H.O. Wang, A fuzzy Lyapunov approach to fuzzy control system design, in Proceedings of the American Control Conference, Arlington, V (2001), pp. 4790–4795
H. Tuan, P. Apkarian, T. Nguyen, T. Narikiyo, Robust mixed H 2–H ∞ filtering of 2D systems. IEEE Trans. Signal Process. 50(7), 1759–1771 (2002)
L. Wu, Z. Wang, H. Gao, C. Wang, Filtering for uncertain 2D discrete systems with state delays. Signal Process. 87(9), 2213–2230 (2007)
L. Wu, P. Shi, H. Gao, C. Wang, H ∞ filtering for 2D Markovian jump systems. Automatica 44(7), 1849–1858 (2008)
L. Wu, Z. Wang, H. Gao, C. Wang, H ∞ and l 2−l 1 filtering for two-dimensional linear parameter-varying systems. Int. J. Robust Nonlinear Control 17(12), 1129–1154 (2007)
Z. Xiang, S. Huang, Stability analysis and stabilization of discrete-time 2D switched systems. Circuits Syst. Signal Process. 32(1), 401–414 (2013)
X. Xie, H. Zhang, Convergent stabilization conditions of discrete-time 2D T-S fuzzy systems via improved homogeneous polynomial techniques. Acta Autom. Sin. 36(9), 1305–1311 (2010)
X. Xie, H. Zhang, Stabilization of discrete-time 2D T-S fuzzy systems based on new relaxed conditions. Acta Autom. Sin. 36(2), 267–273 (2010)
S. Xu, J. Lam, Y. Zou, Z. Lin, W. Paszke, Robust H ∞ filtering for uncertain 2D continuous systems. IEEE Trans. Signal Process. 53(5), 1731–1738 (2005)
S. Xu, J. Lam, Exponential H ∞ filter design for uncertain Takagi–Sugeno fuzzy systems with time delay. J. Eng. Appl. Artif. Intell. 17(6), 645–659 (2004)
H. Xu, Z. Lin, A. Makur, Non-fragile H 2 and H ∞ filter designs for polytopic two-dimensional systems in Roesser model. Multidimens. Syst. Signal Process. 21(3), 255–275 (2010)
F. Yang, Y. Li, Set-membership fuzzy filtering for nonlinear discrete-time systems. IEEE Trans. Syst. Man Cybern., Part B, Cybern. 40(1), 116–124 (2010)
R. Yang, L. Xie, C. Zhang, H 2 and mixed H 2/H ∞ control of two-dimensional systems in Roesser model. Automatica 42(9), 1507–1514 (2006)
S. Zhou, J. Lam, A. Xue, H ∞ filtering of discrete-time fuzzy systems via basis-dependent Lyapunov function approach. Fuzzy Sets Syst. 158(2), 180–193 (2007)
S. Zhou, B. Zhang, W.X. Zheng, Gain-scheduled H ∞ filtering of parameter-varying systems. Int. J. Robust Nonlinear Control 16(8), 397–411 (2006)
S. Zhou, T. Li, Robust stabilization for delayed discrete-time fuzzy systems via basis-dependent Lyapunov-Krasovskii function. Fuzzy Sets Syst. 151(1), 139–153 (2005)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Boukili, B., Hmamed, A., Benzaouia, A. et al. H ∞ Filtering of Two-Dimensional T-S Fuzzy Systems. Circuits Syst Signal Process 33, 1737–1761 (2014). https://doi.org/10.1007/s00034-013-9720-2
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00034-013-9720-2